Program f08krfe
! F08KRF Example Program Text
! Mark 27.3 Release. NAG Copyright 2021.
! .. Use Statements ..
Use nag_library, Only: nag_wp, zgesdd
! .. Implicit None Statement ..
Implicit None
! .. Parameters ..
Integer, Parameter :: nb = 64, nin = 5, nout = 6, &
prerr = 0
! .. Local Scalars ..
Integer :: i, info, lda, ldu, ldvt, lwork, m, n
! .. Local Arrays ..
Complex (Kind=nag_wp), Allocatable :: a(:,:), a_copy(:,:), b(:), u(:,:), &
vt(:,:), work(:)
Complex (Kind=nag_wp) :: dummy(1,1)
Real (Kind=nag_wp), Allocatable :: rwork(:), s(:)
Integer, Allocatable :: iwork(:)
! .. Intrinsic Procedures ..
Intrinsic :: max, min, nint, real
! .. Executable Statements ..
Write (nout,*) 'F08KRF Example Program Results'
Write (nout,*)
! Skip heading in data file
Read (nin,*)
Read (nin,*) m, n
lda = m
ldu = m
ldvt = n
Allocate (a(lda,n),a_copy(m,n),s(m),u(ldu,m),vt(ldvt,n),b(m),rwork((5*m+ &
7)*n),iwork(8*m))
! Read the m by n matrix A from data file
Read (nin,*)(a(i,1:n),i=1,m)
! Read the right hand side of the linear system
Read (nin,*) b(1:m)
a_copy(1:m,1:n) = a(1:m,1:n)
! Use routine workspace query to get optimal workspace.
lwork = -1
! The NAG name equivalent of dgesdd is f08krf
Call zgesdd('A',m,n,a,lda,s,u,ldu,vt,ldvt,dummy,lwork,rwork,iwork,info)
! Make sure that there is enough workspace for block size nb.
lwork = max((2*m+2)*m+2*n+nb*(m+n),nint(real(dummy(1,1))))
Allocate (work(lwork))
! Compute the singular values and left and right singular vectors
! of A.
! The NAG name equivalent of dgesdd is f08krf
Call zgesdd('A',m,n,a,lda,s,u,ldu,vt,ldvt,work,lwork,rwork,iwork,info)
If (info/=0) Then
Write (nout,99999) 'Failure in F08KRF/ZGESDD. INFO =', info
99999 Format (1X,A,I4)
Go To 100
End If
! Print the significant singular values of A
Write (nout,*) 'Singular values of A:'
Write (nout,99998) s(1:min(m,n))
99998 Format (1X,4(3X,F11.4))
If (prerr>0) Then
Call compute_error_bounds(m,n,s)
End If
If (m<n) Then
! Compute V*Inv(S)*U^T * b to get minimum norm solution.
Call compute_minimum_norm(m,n,a_copy,m,u,ldu,vt,ldvt,s,b)
End If
100 Continue
Contains
Subroutine compute_minimum_norm(m,n,a,lda,u,ldu,vt,ldvt,s,b)
! .. Use Statements ..
Use nag_library, Only: dznrm2, zgemv
! .. Implicit None Statement ..
Implicit None
! .. Scalar Arguments ..
Integer, Intent (In) :: lda, ldu, ldvt, m, n
! .. Array Arguments ..
Complex (Kind=nag_wp), Intent (In) :: a(lda,n), u(ldu,m), vt(ldvt,n)
Complex (Kind=nag_wp), Intent (Inout) :: b(m)
Real (Kind=nag_wp), Intent (In) :: s(m)
! .. Local Scalars ..
Complex (Kind=nag_wp) :: alpha, beta
Real (Kind=nag_wp) :: norm
! .. Local Arrays ..
Complex (Kind=nag_wp), Allocatable :: x(:), y(:)
! .. Intrinsic Procedures ..
Intrinsic :: allocated, cmplx
! .. Executable Statements ..
Allocate (x(n),y(m))
! Compute V*Inv(S)*U^H * b to get least squares solution.
! y = U^H b
! The NAG name equivalent of zgemv is f06saf
alpha = cmplx(1.0_nag_wp,0.0_nag_wp,kind=nag_wp)
beta = cmplx(0.0_nag_wp,0.0_nag_wp,kind=nag_wp)
Call zgemv('C',m,m,alpha,u,ldu,b,1,beta,y,1)
y(1:m) = y(1:m)/s(1:m)
! x = V y
Call zgemv('C',m,n,alpha,vt,ldvt,y,1,beta,x,1)
Write (nout,*)
Write (nout,*) 'Minimum norm solution:'
Write (nout,99999) x(1:n)
norm = dznrm2(n,x,1)
Write (nout,*)
Write (nout,*) 'Norm of Solution:'
Write (nout,99998) norm
! Find norm of residual ||b-Ax||, should be zero.
alpha = cmplx(-1.0_nag_wp,0.0_nag_wp,kind=nag_wp)
beta = cmplx(1._nag_wp,0.0_nag_wp,kind=nag_wp)
Call zgemv('N',m,n,alpha,a,lda,x,1,beta,b,1)
norm = dznrm2(m,b,1)
Write (nout,*)
Write (nout,*) 'Norm of Residual:'
Write (nout,99998) norm
If (allocated(x)) Then
Deallocate (x)
End If
If (allocated(y)) Then
Deallocate (y)
End If
99999 Format (4X,'(',F8.4,',',F8.4,')')
99998 Format (4X,F11.4)
End Subroutine compute_minimum_norm
Subroutine compute_error_bounds(m,n,s)
! Error estimates for singular values and vectors is computed
! and printed here.
! .. Use Statements ..
Use nag_library, Only: ddisna, nag_wp, x02ajf
! .. Implicit None Statement ..
Implicit None
! .. Scalar Arguments ..
Integer, Intent (In) :: m, n
! .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: s(m)
! .. Local Scalars ..
Real (Kind=nag_wp) :: eps, serrbd
Integer :: i, info
! .. Local Arrays ..
Real (Kind=nag_wp), Allocatable :: rcondu(:), rcondv(:), uerrbd(:), &
verrbd(:)
! .. Executable Statements ..
Allocate (rcondu(n),rcondv(n),uerrbd(n),verrbd(n))
! Get the machine precision, EPS and compute the approximate
! error bound for the computed singular values. Note that for
! the 2-norm, S(1) = norm(A)
eps = x02ajf()
serrbd = eps*s(1)
! Call DDISNA (F08FLF) to estimate reciprocal condition
! numbers for the singular vectors
Call ddisna('Left',m,n,s,rcondu,info)
Call ddisna('Right',m,n,s,rcondv,info)
! Compute the error estimates for the singular vectors
Do i = 1, n
uerrbd(i) = serrbd/rcondu(i)
verrbd(i) = serrbd/rcondv(i)
End Do
! Print the approximate error bounds for the singular values
! and vectors
Write (nout,*)
Write (nout,*) 'Error estimate for the singular values'
Write (nout,99999) serrbd
Write (nout,*)
Write (nout,*) 'Error estimates for the left singular vectors'
Write (nout,99999) uerrbd(1:n)
Write (nout,*)
Write (nout,*) 'Error estimates for the right singular vectors'
Write (nout,99999) verrbd(1:n)
99999 Format (4X,1P,6E11.1)
End Subroutine compute_error_bounds
End Program f08krfe